1178edo: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|1178}}
{{ED intro}}


== Theory ==
== Theory ==
1178edo is a very strong 19-limit system, and is a [[zeta edo|zeta peak, integral and gap edo]]. It is also [[consistency|distinctly consistent]] through to the [[21-odd-limit]], and is the first edo past [[742edo|742]] with a lower 19-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]]. A basis for its 19-limit [[comma]]s is [[2500/2499]], [[3025/3024]], 3250/3249, 4200/4199, [[4375/4374]], [[4914/4913]] and 5985/5984. It [[support]]s and provides a great tuning for [[semihemienneadecal]].  
1178edo is a very strong 19-limit system, and is a [[zeta edo|zeta peak, integral and gap edo]]. It is also [[consistency|distinctly consistent]] through to the [[21-odd-limit]], and is the first edo past [[742edo|742]] with a lower 19-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]]. A [[comma basis|basis]] for its 19-limit [[comma]]s consists of [[2500/2499]], [[3025/3024]], 3250/3249, 4200/4199, [[4225/4224]], [[4375/4374]], and [[4914/4913]]. It [[support]]s and provides a great tuning for [[semihemienneadecal]].  


=== Prime harmonics ===
=== Prime harmonics ===
Line 9: Line 9:


=== Subsets and supersets ===
=== Subsets and supersets ===
Since 1178 = {{factorization|1178}}, 1178edo is notable for containing both 19 and 31. Its subset edos are {{EDOs| 2, 19, 31, 38, 62, and 589 }}.
Since 1178 factors into {{factorization|1178}}, 1178edo is notable for containing both 19 and 31. Its subset edos are {{EDOs| 2, 19, 31, 38, 62, and 589 }}.


== Regular temperament properties ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" |[[Subgroup]]
! rowspan="2" |[[Comma list|Comma List]]
! rowspan="2" |[[Mapping]]
! rowspan="2" |Optimal<br>8ve Stretch (¢)
! colspan="2" |Tuning Error
|-
![[TE error|Absolute]] (¢)
![[TE simple badness|Relative]] (%)
|-
|-
|2.3
! rowspan="2" | [[Subgroup]]
|{{monzo|-1867 1178}}
! rowspan="2" | [[Comma list]]
|{{val|1178 1867}}
! rowspan="2" | [[Mapping]]
| 0.0276
! rowspan="2" | Optimal<br />8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -1867 1178 }}
| {{mapping| 1178 1867 }}
| +0.0276
| 0.0276
| 0.0276
| 2.71
| 2.71
|-
|-
|2.3.5
| 2.3.5
|{{monzo|-14 -19-19}}, {{monzo|-99 61 1}}
| {{monzo| -14 -19-19 }}, {{monzo| -99 61 1 }}
|{{val|1178 1867 2735}}
| {{mapping| 1178 1867 2735 }}
| 0.0522
| +0.0522
| 0.0415
| 0.0415
| 4.07
| 4.07
|-
|-
|2.3.5.7
| 2.3.5.7
|4375/4374, 703125/702464, 2460375/2458624
| 4375/4374, 703125/702464, {{monzo| -52 -5 -2 23 }}
|{{val|1178 1867 2735 3307}}
| {{mapping| 1178 1867 2735 3307 }}
| 0.0450
| +0.0450
| 0.0380
| 0.0380
| 3.73
| 3.73
|-
|-
|2.3.5.7.11
| 2.3.5.7.11
|3025/3024, 4375/4374, 9801/9800, 43923/43904
| 3025/3024, 4375/4374, 234375/234256, {{monzo| -27 3 -4 10 1 }}
|{{val|1178 1867 2735 3307 4075}}
| {{mapping| 1178 1867 2735 3307 4075 }}
| 0.0484
| +0.0484
| 0.0347
| 0.0347
| 3.41
| 3.41
|-
|-
|2.3.5.7.11.13
| 2.3.5.7.11.13
|3025/3024, 4225/4224, 9801/9800, 91125/91091, 1664000/1663893
| 3025/3024, 4225/4224, 4375/4374, 78125/78078, 1664000/1663893
|{{val|1178 1867 2735 3307 4075 4359}}
| {{mapping| 1178 1867 2735 3307 4075 4359 }}
| 0.0457
| +0.0457
| 0.0322
| 0.0322
| 3.16
| 3.16
|-
|-
|2.3.5.7.11.13.17
| 2.3.5.7.11.13.17
|3025/3024, 4225/4224, 5832/5831, 9801/9800, 14875/14872, 31213/31212
| 2500/2499, 3025/3024, 4225/4224, 4375/4374, 4914/4913, 14875/14872
|{{val|1178 1867 2735 3307 4075 4359 4815}}
| {{mapping| 1178 1867 2735 3307 4075 4359 4815 }}
| 0.0403
| +0.0403
| 0.0327
| 0.0327
| 3.21
| 3.21
|-
|-
|2.3.5.7.11.13.17.19
| 2.3.5.7.11.13.17.19
|4200/4199, 5985/5984, 9801/9800, 10830/10829, 12636/12635, 14365/14364, 23409/23408
| 2500/2499, 3025/3024, 3250/3249, 4200/4199, 4225/4224, 4375/4374, 4914/4913
|{{val|1178 1867 2735 3307 4075 4359 4815 5004}}
| {{mapping| 1178 1867 2735 3307 4075 4359 4815 5004 }}
| 0.0370
| +0.0370
| 0.0318
| 0.0318
| 3.12
| 3.12
|}
|}
=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
! Periods<br>per 8ve
|-
! Generator<br>(reduced)
! Periods<br />per 8ve
! Cents<br>(reduced)
! Generator*
! Associated<br>ratio
! Cents*
! Associated<br />ratio*
! Temperaments
! Temperaments
|-
|-
|1
| 1
|337\1178
| 337\1178
|343.29
| 343.29
|8000/6561
| 8000/6561
|[[Raider]]
| [[Raider]]
|-
| 19
| 489\1178<br />(7\1178)
| 498.13<br />(7.13)
| 4/3<br />(225/224)
| [[Enneadecal]]
|-
|-
|19
| 31
|489\1178<br>(7\1178)
| 581\1178<br />(11\1178)
|498.13<br>(7.13)
| 591.851<br />(11.205)
|4/3<br>(225/224)
| 936/665<br />(?)
|[[Enneadecal]]
| [[31st-octave temperaments#217 & 1178|217 & 1178]]
|-
|-
|38
| 38
|489\1178<br>(7\1178)
| 260\1178<br />(12\1178)
|498.13<br>(7.13)
| 264.86<br />(12.22)
|4/3<br>(225/224)
| 500/429<br />(144/143)
|[[Hemienneadecal]]
| [[Semihemienneadecal]]
|-
| 38
| 489\1178<br />(7\1178)
| 498.13<br />(7.13)
| 4/3<br />(225/224)
| [[Hemienneadecal]]
|}
|}
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct


== Music ==
== Music ==
; '''[[Eliora]]'''
; [[Eliora]]
* [https://www.youtube.com/watch?v=c9e7MTsIDc4 Listening] (2023) - 217 & 1178 and enneadecal in 1178edo tuning
* [https://www.youtube.com/watch?v=c9e7MTsIDc4 ''Listening''] (2023) – {{nowrap|217 &amp; 1178}} and enneadecal in 1178edo tuning
 
[[Category:Enneadecal]]
[[Category:Enneadecal]]
[[Category:Hemienneadecal]]
[[Category:Hemienneadecal]]
[[Category:Listen]]
[[Category:Semihemienneadecal]]
[[Category:Semihemienneadecal]]
[[Category:Listen]]

Latest revision as of 16:53, 18 February 2025

← 1177edo 1178edo 1179edo →
Prime factorization 2 × 19 × 31
Step size 1.01868 ¢ 
Fifth 689\1178 (701.868 ¢)
Semitones (A1:m2) 111:89 (113.1 ¢ : 90.66 ¢)
Consistency limit 21
Distinct consistency limit 21

1178 equal divisions of the octave (abbreviated 1178edo or 1178ed2), also called 1178-tone equal temperament (1178tet) or 1178 equal temperament (1178et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 1178 equal parts of about 1.02 ¢ each. Each step represents a frequency ratio of 21/1178, or the 1178th root of 2.

Theory

1178edo is a very strong 19-limit system, and is a zeta peak, integral and gap edo. It is also distinctly consistent through to the 21-odd-limit, and is the first edo past 742 with a lower 19-limit relative error. A basis for its 19-limit commas consists of 2500/2499, 3025/3024, 3250/3249, 4200/4199, 4225/4224, 4375/4374, and 4914/4913. It supports and provides a great tuning for semihemienneadecal.

Prime harmonics

Approximation of prime harmonics in 1178edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.000 -0.087 -0.236 -0.065 -0.214 -0.120 -0.032 -0.060 +0.249 +0.304 -0.044
Relative (%) +0.0 -8.6 -23.1 -6.4 -21.0 -11.8 -3.1 -5.9 +24.4 +29.8 -4.3
Steps
(reduced) 		1178
(0) 		1867
(689) 		2735
(379) 		3307
(951) 		4075
(541) 		4359
(825) 		4815
(103) 		5004
(292) 		5329
(617) 		5723
(1011) 		5836
(1124)

Subsets and supersets

Since 1178 factors into 2 × 19 × 31, 1178edo is notable for containing both 19 and 31. Its subset edos are 2, 19, 31, 38, 62, and 589.

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [-1867 1178⟩ [⟨1178 1867]] +0.0276 0.0276 2.71
2.3.5 [-14 -19-19⟩, [-99 61 1⟩ [⟨1178 1867 2735]] +0.0522 0.0415 4.07
2.3.5.7 4375/4374, 703125/702464, [-52 -5 -2 23⟩ [⟨1178 1867 2735 3307]] +0.0450 0.0380 3.73
2.3.5.7.11 3025/3024, 4375/4374, 234375/234256, [-27 3 -4 10 1⟩ [⟨1178 1867 2735 3307 4075]] +0.0484 0.0347 3.41
2.3.5.7.11.13 3025/3024, 4225/4224, 4375/4374, 78125/78078, 1664000/1663893 [⟨1178 1867 2735 3307 4075 4359]] +0.0457 0.0322 3.16
2.3.5.7.11.13.17 2500/2499, 3025/3024, 4225/4224, 4375/4374, 4914/4913, 14875/14872 [⟨1178 1867 2735 3307 4075 4359 4815]] +0.0403 0.0327 3.21
2.3.5.7.11.13.17.19 2500/2499, 3025/3024, 3250/3249, 4200/4199, 4225/4224, 4375/4374, 4914/4913 [⟨1178 1867 2735 3307 4075 4359 4815 5004]] +0.0370 0.0318 3.12

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperaments
1 337\1178 343.29 8000/6561 Raider
19 489\1178
(7\1178) 		498.13
(7.13) 		4/3
(225/224) 		Enneadecal
31 581\1178
(11\1178) 		591.851
(11.205) 		936/665
(?) 		217 & 1178
38 260\1178
(12\1178) 		264.86
(12.22) 		500/429
(144/143) 		Semihemienneadecal
38 489\1178
(7\1178) 		498.13
(7.13) 		4/3
(225/224) 		Hemienneadecal

* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct

Music

Eliora
  • Listening (2023) – 217 & 1178 and enneadecal in 1178edo tuning
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